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Solving and Graphing Multistep Inequalities - Problem 3

Teacher/Instructor Alissa Fong
Alissa Fong

MA, Stanford University
Teaching in the San Francisco Bay Area

Alissa is currently a teacher in the San Francisco Bay Area and Brightstorm users love her clear, concise explanations of tough concepts

In multi-step inequalities, start by simplifying both sides of the inequality sign, as needed. This means to get rid of any parenthesis by distributing, combine any like terms, and so forth. When using the distributive property, it is very important to distribute the sign of the number as well. If there is a negative number outside of the parenthesis, make sure you are multiplying all the terms inside by a negative number. After the inequality has been simplified, use inverse operations to solve for the unknown variable. Remember that solving an inequality and finding the value of the variable requires "undoing" what has been done to the variable. Work in the reverse order of PEMDAS. For example, if the variable is being multiplied by 2 then subtracted by 4, start by adding 4 to both sides, then dividing by 2 to both sides. Don't forget to change the inequality sign whenever you multiply or divide both sides by a negative value. Once you solve the inequality, graph it on a number line.

This is going to be a more difficult multistep inequality, because I'm going to have to distribute, and combine like terms, in order to get x isolated by itself.

So the first thing I want to do on the right hand side is just simplify. In order to do that I'm going that I'm going to distribute this 3. This stuff all stays the same, then I'm going to have +3x minus 12. Now to continue simplifying. I need to combine like terms. -7x plus 3 more x's is going to be -4x.

Next I want to get my x's all alone. Right now my x is attached to this -12 piece. So in order to get rid of it, I'm going to add 12 to both sides. So that I'll have 10 is greater than -4x. I'm almost done solving. I want to know what regular x is, not what -4x is.

This means -4 times x, and the opposite of multiplying is to divide. So I'm going to be dividing both sides by -4. Since I divided by a negative value, this sine is going to change directions. I'm going to flip it. Instead of being a greater than, it's going to become a less than. 10 over -4 is less than x. That's super important. It's a really common mistake students make. They go through, and they do all the correct. All of this stuff correctly, and then they blow it, because they forget to change the direction.

Last, but not least I'm going to reduce that fraction. I'm going to step this way a little bit, move my problem over here. 10 over -4, if I divide top, and bottom by 2, I'll get 5/2 with a negative like that is less than x.

If this doesn't make sense in your brain, if you like to have your variable on the left hand side, you could also write it like this. Just be really careful that the alligator, if you think of this as an alligator, he want to eat the x. X is the bigger one. So make sure when you redraw it, that x is still getting eaten by the alligator. X is the bigger value.

When I draw it, I want to mark numbers that are greater than -5/2. -5/2 is the same thing as -2½, if you like mixed numbers instead of fractions. So when I draw it, I'm going to approximate -2½ about right there. I'm going to make it an open circle. I want to mark numbers that are larger than that. X needs to be larger than -5/2. So my solutions are numbers that are larger than -5/2 or -2½.

This is a problem where you had to be really careful with the negative sign, and the flip. That's the place that people make a lot of errors. So when you're doing your homework, and you're checking in the back of the book, if you notice that, what you get is just a tiny bit different, from what's in the back of the book, my guess is that, you forgot to flip this inequality sign; if you multiplied or divided by a negative number.

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